Topological derivatives for thermo-mechanical semi-coupled system

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The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. According to the literature, the topological derivative has been fully developed for a wide range of one single physical phenomenon modeled by partial differential equations. In addition, up to our knowledge, the topological asymptotic analysis associated to multi-physics problems has so far not been re- ported in the literature. In this work, we present the topological asymptotic analysis for the total potential mechanical energy associated to a thermo-mechanical system, when a small circular inclusion is introduced at an arbitrary point of the domain. In particular, we con- sider the linear elasticity system (modeled by the Navier equation) coupled with the steady- state heat conduction problem (modeled by the Laplace equation). The mechanical coupling term comes out from the thermal stress induced by the temperature field. Since this term is non-local, we introduce a non-standard adjoint state, which allows to obtain a closed form for the topological derivative. Finally, we provide a full mathematical justification for the de- rived formulas and develop precise estimates for the remainders of the topological asymptotic expansion.